18.783 Elliptic Curves Lecture 1 - MIT Mathematics

18.783 Elliptic Curves

Lecture 1

Andrew Sutherland

February 8, 2017

What is an elliptic curve?

The equation

x2

a2

+

y2

b2

= 1 defines an ellipse.

An ellipse, like all conic sections, is a curve of genus 0.

It is not an elliptic curve. Elliptic curves have genus 1.

The area of this ellipse is ¦Ðab. What is its circumference?

The circumference of an ellipse

p

Let y = f (x) = b 1¡Ì? x2 /a2 .

Then f 0 (x) = ?rx/ a2 ? x2 , where r = b/a < 1.

Applying the arc length formula, the circumference is

Z ap

Z ap

4

1 + f 0 (x)2 dx = 4

1 + r2 x2 /(a2 ? x2 ) dx

0

0

With the substitution x = at this becomes

Z 1r

1 ? e2 t2

4a

dt,

1 ? t2

0

¡Ì

where e = 1 ? r2 is the eccentricity of the ellipse.

This is an elliptic integral. The integrand u(t) satisfies

u2 (1 ? t2 ) = 1 ? e2 t2 .

This equation defines an elliptic curve.

An elliptic curve over the real numbers

With a suitable change of variables, every elliptic curve with real

coefficients can be put in the standard form

y 2 = x3 + Ax + B,

for some constants A and B. Below is an example of such a curve.

y 2 = x3 ? 4x + 6

over R

An elliptic curve over a finite field

y 2 = x3 ? 4x + 6

over F197

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